"Use Excel to approximate dF/dt=-0.1F+70, F(0)=0 to generate approximations for F at t=1,2 and 4 using step size 0.1. Explain whether these approximation are greater than or less than the exact values. Determine whether the shape of the solution curve is increasing, decreasing, concave up or concave down based on the data alone. Explain."
ynext = ynow + (1/6)(k1 +2k2 + 2k3 + k4)
k1= Δx*f'(xnow, ynow)
k2= Δx*f'(xnow+.5Δx,ynow +.5k1)
k3= Δx*f'(xnow+.5Δx,ynow +.5k2)
I found the exact solution to be F(t)=700-700*exp(-.1t)
The Attempt at a Solution
i've attached my excel file.
Since Runge-Kutta is al inear technique and I observed all the slopes in RK to be decreasing I predicted that the method would give an over estimate. However, comparing my estimate with the exact solution shows that I infact get an under-approximation. I am completely lost as to why this occurs and am looking for some explanation. I do not feel i am able to say anything about the concavity given my results that the decreasing rates (f'1, f'2 et cetera on my excel) gave an underestimation.
futher, how exactly does putting weight on the k2 and k3 terms affect the approximation?